Estimation of truncated regression models using the Quadratic Mode Estimator (QME)

Estimation of linear regression models with truncated response variables (fixed truncation point), using the Quadratic Mode Estimator (QME) (Lee 1993 and Laitila 2001)

qme(formula, data, point = 0, direction = "left", cval = "ml", 
  const = 1, beta = "ml", covar = FALSE, na.action, ...)
## S3 method for class 'qme':
print(x, digits = max(3, getOption("digits") - 3), ...)
## S3 method for class 'qme':
summary(object, level=0.95, ...)
## S3 method for class 'summary.qme':
print(x, digits = max(3, getOption("digits") - 3), ...)
## S3 method for class 'qme':
## S3 method for class 'qme':
## S3 method for class 'qme':
## S3 method for class 'qme':
x, object
an object of class "qme"
a symbolic description of the model to be estimated
an optional data frame
the value of truncation (the default is 0)
the direction of truncation, either "left" (the default) or "right"
the threshold value to be used when trimming the conditional density of the errors. The default is "ml" meaning that the estimated residual standard deviation from a maximum likelihood model for truncated regression, fitted using
a number that can be used to alter the size of the threshold value. const=0.5 would give a threshold value that is half the original size. The default value is 1.
the method of determining the starting values of the regression coefficients (See Details for more information):
  • The default method is"ml", meaning that the estimated regression coefficients from fitting a maximum likelihood model fo
logical. Indicates whether or not the covariance matrix should be estimated. If TRUE the covariance matrix is estimated using bootstrap, as described in Karlsson (2004). The default number of replicates is 2000 but this can be adjusted (see a
a function which indicates what should happen when the data contain NAs.
the number of digits to be printed
the desired level of confidence, for confidence intervals provided by summary.qme. A number between 0 and 1. The default value is 0.95.
additional arguments. For qme the number of bootstrap replicates can be adjusted by setting R=the desired number of replicates. Also the control argument of optim can be

Finds the QME estimates of the regression coefficients by maximizing the objective function described in Lee (1993) wrt the vector of regression coefficients. The maximization is performed by optim using the "Nelder--Mead" method. The maximum number of iterations is set at 2000, but this can be adjusted by setting control=list(maxit=...) (for more information see the documentation for optim). The starting values of the regression coefficients can have a great impact on the result of the maximization. For this reason it is recommended to use one of the methods for generating these rather than supplying the values manually, unless one is confident that one has a good idea of what the starting values should be. For more detailed information see Karlsson and Lindmark (2014).


  • qme returns an object of class "qme". The function summary prints a summary of the results, including two types of confidence intervals (normal approximation and percentile method). The generic accessor functions coef, fitted, residuals and vcov extract various useful features of the value returned by qme An object of class "qme", a list with elements:
  • coefficientsthe named vector of coefficients
  • startcoefthe starting values of the regression coefficients used by optim
  • cvalinformation about the threshold value used. The method and constant value used and the resulting threshold value.
  • valuethe value of the objective function corresponding to coefficients
  • countsnumber of iterations used by optim. See the documentation for optim for further details
  • convergencefrom optim. An integer code. 0 indicates successful completion. Possible error codes are 1 indicating that the iteration limit maxit had been reached. 10 indicating degeneracy of the Nelder--Mead simplex.
  • messagefrom optim. A character string giving any additional information returned by the optimizer, or NULL.
  • residualsthe residuals of the model
  • fitted.valuesthe fitted values
  • df.residualthe residual degrees of freedom
  • callthe matched call
  • covarianceif covar=TRUE, the estimated covariance matrix
  • Rif covar=TRUE, the number of bootstrap replicates
  • bootreplif covar=TRUE, the bootstrap replicates


Karlsson, M. (2004) Finite sample properties of the QME, Communications in Statistics - Simulation and Computation, 5, pp 567--583 Karlsson, M., Lindmark, A. (2014) truncSP: An R Package for Estimation of Semi-Parametric Truncated Linear Regression Models, Journal of Statistical Software, 57(14), pp 1--19, http://www.jstatsoft.org/v57/i14/ Laitila, T. (2001) Properties of the QME under asymmetrically distributed disturbances, Statistics & Probability Letters, 52, pp 347--352 Lee, M. (1993) Quadratic mode regression, Journal of Econometrics, 57, pp 1-19 Lee, M. & Kim, H. (1998) Semiparametric econometric estimators for a truncated regression model: a review with an extension, Statistica Neerlandica, 52(2), pp 200--225

See Also

qme.fit, the function that does the actual fitting lt, for estimation of models with truncated response variables using the LT estimator stls, for estimation of models with truncated response variables using the STLS estimator truncreg for estimating models with truncated response variables by maximum likelihood, assuming Gaussian errors

  • qme
  • print,qme-method
  • summary,qme-method
  • print,summary.qme-method
  • coef,qme-method
  • vcov,qme-method
  • residuals,qme-method
  • fitted,qme-method
##Simulate a data.frame (model with asymmetrically distributed errors)
n <- 10000
x1 <- runif(n,0,10)
x2 <- runif(n,0,10)
x3 <- runif(n,-5,5)
eps <- rexp(n,0.2)- 5
y <- 2-2*x1+x2+2*x3+eps
d <- data.frame(y=y,x1=x1,x2=x2,x3=x3)
##Use a truncated subsample
dtrunc <- subset(d, y>0)

##Use qme to consistently estimate the slope parameters
qme(y~x1+x2+x3, dtrunc, point=0, direction="left", cval="ml", const=1, 
   beta="ml", covar=FALSE)
##Example using data "PM10trunc"

qmepm10 <- qme(PM10~cars+temp+wind.speed+temp.diff+wind.dir+hour+day, 
   data=PM10trunc, point=2, control=list(maxit=4500))

Documentation reproduced from package truncSP, version 1.2.2, License: GPL (>= 2)

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